CHAPTER 2 Describing Distributions with Numbers Numerical measures are often used to describe distributions. Select Data h Data Analysis h Descriptive Statistics from the menu to obtain descriptive statistics. Enter the input range for the data. If the data includes a label in the first row, check the appropriate box. Specify where the output will appear, check the box next to Summary Statistics, and click OK in the following dialog box. 14 Chapter 2: Describing Distributions with Numbers The command summarizes several different measures of both the center and spread of a distribution. The command prints the following: Mean, Standard Error, Median, Mode, Standard Deviation, Sample Variance, Kurtosis, Skewness, Range, Minimum, Maximum, Sum, and Count, for each column specified. Count is the number of actual values in the column (missing values are not counted). Mean is the average of the values. To find the median, the data first must be ordered. If N is odd, the median is the value in the middle. If N is even, the median is the average of the two middle values. StDev is the standard deviation computed as StDev =
∑ (x − x)
i
N −1
2
Standard Error is the standard error of the mean. It is calculated as StDev N . The same results can be obtained individually using functions in Excel. For example, typing =AVERAGE(A2:A61) into a cell gives the mean, =STDEV(A2:A61) gives the standard deviation, and =COUNT(A2:A61) gives the sample size. In addition, we can obtain the quartiles needed for the five‐number summary using the QUARTILE function. If you click on the Insert Function, , you can search for the appropriate function as shown below. Chapter 2: Describing Distributions with Numbers 15 As shown below, you can obtain the function either by typing the formula into an empty cell, clicking on an empty cell and then typing the formula into the formula bar to the right of the Insert Function button, or by filling in a dialog box. Either way, you must specify the Array that holds the data and then the Quart, where Quart = 0 is the minimum value, 1 is the first quartile, 2 is the median, 3 is the third quartile, and 4 is the maximum. Excel doesn’t use exactly the same algorithm to calculate quartiles as your textbook, so minor differences in results will sometimes occur. 16 Chapter 2: Describing Distributions with Numbers The five‐number summary consisting of the median, quartiles, and minimum and maximum values provides a quick overall description of a distribution. To do boxplots, you must use the Add‐Ins > WHFStat > Graphs > Boxplot. This tool will also give you the five number summary. Mean To find the mean of a set of numbers we will use the data from Exercise 2.3 Enter the data then choose Formulas > Insert function > Average. Fill in the dialog box with the data range. Chapter 2: Describing Distributions with Numbers 17 You can see the result on the bottom left of the screen. Mean = 31.25 18 Chapter 2: Describing Distributions with Numbers To find the median, choose Formulas > Insert function > Median. Again, you can see the result in the bottom left corner: Median = 22.5 The mean is larger due to the right‐skew of the distribution. The Five Number Summary and BoxPlot To illustrate this we will use Exercise 2.29 Enter the data in columns A, B, C. Choose Add‐Ins > WHFStat > Graphs > Boxplot from the Menu. Chapter 2: Describing Distributions with Numbers 19 Highlight all three columns to get side‐by‐side boxplots. The Five Number summary will be part of the output. If only descriptive statistics are needed, and not a box plot Select Data h Data Analysis h Descriptive Statistics . This is illustrated in Exercise 2.37. Enter the data into columns A and B. 20 Chapter 2: Describing Distributions with Numbers The mean is 8.4%. The smaller states have too much weight in this average. Selected exercises. Try the following exercises using Excel. 2.1 Pulling wood apart 2.3 New York travel times 2.5 Food oils 2.35 Guinea pig survival times 2.37 Thinking about means 2.43 Athletes Salaries 2.45 Do good smells bring good business? 2.47 Compressing Soil 2.49 Older Americans 2.51 Athlete’s Salaries